Closed form solution for the surface area, the capacitance and the demagnetizing factors of the ellipsoid

TL;DR

This paper presents exact closed-form solutions for the surface area, capacitance, and demagnetizing factors of ellipsoids using hypergeometric functions, with implications across physics, biology, and space science.

Contribution

It introduces novel closed-form formulas for key ellipsoid properties expressed via hypergeometric functions, extending previous approximate or numerical methods.

Findings

Exact formulas for surface area, capacitance, and demagnetizing factors derived.

Results agree with known solutions for special cases like spheres and spheroids.

Potential applications in various scientific fields discussed.

Abstract

We derive the closed form solutions for the surface area, the capacitance and the demagnetizing factors of the ellipsoid immersed in the Euclidean space R^3. The exact solutions for the above geometrical and physical properties of the ellipsoid are expressed elegantly in terms of the generalized hypergeometric functions of Appell of two variables. Various limiting cases of the theorems of the exact solution for the surface area, the demagnetizing factors and the capacitance of the ellipsoid are derived, which agree with known solutions for the prolate and oblate spheroids and the sphere. Possible applications of the results achieved, in various fields of science, such as in physics, biology and space science are briefly discussed.